2 Answers
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Yes, you can definitely damage the rim and tyre. But normally you just pinch the tube. The little dents you make have sharp edges which are not good for the tyre, and you might possibly poke your tool through the sidewall, but it's extremely unlikely that you will break the bead of the tyre. You might even crack the rim by excerting too much force (this is normally a problem for people who resort to metal tools after breaking plastic tyre levers). But if you're using that much force, you are doing it wrong.

The "magic" technique bike mechanics use for removing tyres is to pick one point on the rim, pull the tyre out at that point, then run your other hand round pushing the bead into the slight depression in the middle of the rim. That gives you extra slack at the hand that's pulling out, making it easier to get the tyre off. It also makes sure the rim isn't stuck to the tyre at any point, and makes sure the bead is disengaged from the hook of the rim. I usually use a tyre lever for that so that any rough spots or objects embedded in the tyre don't end up in my hand.

Look at a rim cross section to see what I mean:

Two things to look at: the hooks that "grab" the bead to hold the tyre in place, and the slight curve that creates a low spot for the bead to sit in while you're getting the tyre off. Done this way I often don't need tyre levers except for running one round the rim to loosen the tyre.

The danger of not using the right tools are that you may scratch or ding the rim, causing further problems later on, or pinch the tube, which will result in the tire remaining flat after you fix the original hole. Bike tire levers are very cheap, you can buy a nice set of the Park ones for about 3 bucks.

Technique... Use the tire lever only for dismounting the tire. Push the stem in a bit, and then push the rim in as far as you can at that point. Then, go to the opposite side of the rim and slide the tire tool in carefully, avoiding pinching the tube.

On most tires, you don't have to actually pry the tire loose, just slide the tool from side to side a bit till you can work it off by hand. Normally, you don't have to totally remove the tire to fix a flat, but you must identify and remove whatever caused it. You should be able to remount the tire with your hands alone, as it's easiest to pinch the tube when trying to use the tool to do this.

I find that skinnier tires are harder to remove without a lever; I almost always end up doing more work to loosen road tires than wider ones -- say, cruiser or offroad tires. (I can usually use the lever to just get those started, then just use my fingers.)
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Neil FeinMay 5 '11 at 21:55

I apologize for my poor English. I meant tire instead of wheel and tube instead of tire.
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Theta30May 5 '11 at 22:47

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@Neil: that's because with skinny tyres the tube is more effective at pushing the bead back to the outside of the rim. And the rims are often flatter, so you don't gain as much by pushing the bead into the centre.
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МסžMay 6 '11 at 4:15

@Mielu - Hold on, not sure I get this -- you're concerned with damaging the tube when removing it, or the tire? Whatever the case, please feel free to edit your answer to make it more accurate.
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Neil FeinMay 6 '11 at 4:39